How to UV unwrap simple plane with very short one edge?
Unfortunately, I don't believe there is any way to do this in blender via UV unwrapping.

The way all 3D programs calculate 4 sided quads is to split them into triangles.

As you scale it, blender is automatically triangulating the four sided quad so it creates a diagonal split along the face in a preferred angle depending on which direction you move the vertices (it changes depending on the algorithm blender uses). 

Therefore one triangle will maintain its scale and the other will be distorted as you scale the bottom edge of one of the triangles (which don't appear to be visible until you start scaling because there is no visible edge separating them).

I suggest two methods. You could either bake your image using your subdivided version to use for your image texture for your 4 sided quad or you could use GIMP or photoshop and morph the image along the bottom edge, using the tools there and you could export your UV map template as another layer to use to help line it up.

Like this:



Blend file:
https://www.dropbox.com/s/rmifrinjqmc34ba/UV%20plane%20stretching.blend?dl=0